Nonlinear Schrödinger–Helmholtz equation as numerical regularization of the nonlinear Schrödinger equation

Abstract

A regularized α-system of the nonlinear Schrödinger (NLS) equation with 2σ nonlinear power in dimension N is studied. We prove short time existence and uniqueness of solution in the case . And in the case 1 ⩽ σ < 3 (when N = 1) or in the case (when N > 1) we show global in time existence of solutions. When α → 0+, the solutions of this regularized system will converge to the solutions of the classical NLS in the appropriate range when the latter exists. Consequently, we propose this regularized system as a numerical regularization to shed light on the profile of the blow-up solutions of the original NLS equation in the range , and in particular for the classical critical case N = 2, σ = 1. Following the modulation theory, we derive the reduced system of ordinary differential equations for the Schrödinger–Helmholtz (SH) system. We observe that the reduced equations for this SH system are more complicated than the equations of some other perturbation regularizations of the classical NLS equation. The detailed analysis of the reduced system on how the regularization prevents singularity formation will be presented in a forthcoming paper.